The Logical Geometry of Russell’s Theory of Definite Descriptions Lorenz Demey∗ Abstract This paper studies Russell’s theory of definite descriptions (TDD) from the per- spective of logical geometry, i.e. by focusing on the various Aristotelian diagrams it gives rise to. Russell analyzed sentences of the form ‘the A is B’ in terms of existence, uniqueness and universality conditions. I first show that each definite de- scription gives rise to four distinct formulas (depending on negation scope), which jointly constitute a classical square of opposition. Next, I discuss the interplay be- tween the Aristotelian square for TDD and that for the categorical statements. After arguing that the latter is already implicitly present in the former, I integrate both into a single Aristotelian diagram, viz. a so-called ‘Buridan octagon’. Finally, I study the exact role of the existence and uniqueness conditions within TDD, by introducing two new logical systems based on these conditions, and showing that this has drastic consequences for the aforementioned Buridan octagon. 1 Introduction Bertrand Russell was notoriously critical of Aristotle and his followers, writing, for ex- ample, that “Aristotle, in spite of his reputation, is full of absurdities” (Russell, 1950, p. 99), and that “[t]hroughout modern times, practically every advance in science, in logic, or in philosophy has had to be made in the teeth of opposition from Aristotle’s disciples” (Russell, 1946, p. 237). Some of his most severe criticisms were directed at Aristotle’s logical theories, which he considered “wholly false, with the exception of the formal theory of the syllogism, which is unimportant” (Russell, 1946, p. 237). One of Russell’s objections against Aristotelian logic concerns its assumption of existential im- port, which would allow one to erroneously infer the (false) conclusion ‘some mountains are golden’ from the (true) premises ‘all golden mountains are mountains’ and ‘all golden mountains are golden’ (Russell, 1946, p. 232). ∗The author holds a postdoctoral fellowship of the Research Foundation–Flanders (FWO). Thanks to Elise Frketich, Jan Heylen, Stephen Neale, Hans Smessaert and Margaux Smets for their valuable feedback on earlier versions of this paper. Also thanks to the audiences of the Center for Logic and Analytic Philos- ophy seminar (Leuven, 15 November 2016), the Logic and Epistemology Research Colloquium (Bochum, 13 July 2017) and the 4th Philosophy of Language and Mind conference (Bochum, 21 – 23 September 2017), where I presented earlier versions of this paper. The work described in this paper was partially carried out during research stays with the Faculty of Philosophy at the University of Oxford (Spring 2017) and at the Institut fur¨ Philosophie II at the Ruhr-Universitat¨ Bochum (Summer 2017). 1 Throughout the history of Aristotelian logic, the square of opposition has been used to visualize the relations of contradiction, (sub)contrariety and subalternation holding between the four categorical statements. In the past few years, there has been a revived interest in the square and its various extensions and reinterpretations, mostly under the label of logical geometry (Smessaert and Demey, 2014a, 2015). Although Russell does not seem to have discussed the square in any detail (Jager, 1972, p. 144), his criticisms of syllogistics are highly relevant here, too: if the assumption of existential import is no longer made, then the categorical square loses its (sub)contrarieties and subalternations. It seems plausible that the particular severity of Russell’s opposition to the tradition of Aristotelian logic is at least partially due to the fact that some of his own most important philosophical achievements went against exactly this tradition. Consider, for example, his statement that “[e]ven at the present day, all Catholic teachers of philosophy and many others still obstinately reject the discoveries of modern logic, and adhere with a strange tenacity to a system which is as definitely antiquated as Ptolemaic astronomy” (Russell, 1946, p. 230).1 The “discoveries of modern logic” that Russell had in mind here probably include his logicist project (Whitehead and Russell, 1910, 1912, 1913) and his theory of definite descriptions (Russell, 1905). Especially the latter has been extremely influential in logic, philosophy of language and linguistics (Neale, 2005), and Russell and his contemporaries were already well aware of its significance. For example, Frank P. Ramsey praised Russell’s theory of definite descriptions as a “paradigm of philosophy” (1931, p. 263), while Russell himself claimed On Denoting to be “his finest philosophical essay” (1956, p. 39). In this paper I will argue that despite these apparent tensions, there exists a fruitful interaction between Aristotelian diagrams (as studied in logical geometry) and Russell’s theory of definite descriptions (henceforth abbreviated as ‘TDD’). Throughout the paper it will be emphasized that this interaction has advantages for both parties involved. On the one hand, the systematic construction of Aristotelian diagrams for TDD will lead us to consider various formulas involving definite descriptions (and logical equiva- lences between such formulas) that have not been investigated in any detail in the litera- ture on TDD so far. Furthermore, these Aristotelian diagrams yield a natural connection with some linguistic-cognitive considerations that are highly relevant for TDD, e.g. con- cerning the nature of natural language negation. Finally, the use of some well-chosen Aristotelian diagrams can be of great help in explaining the features and consequences of TDD, which can be particularly relevant in educational contexts involving students who do not have a strong background in formal logic, but do have extensive knowledge of Aristotelian syllogistics.2 On the other hand, TDD is a rich source of philosophically interesting decorations for a wide variety of Aristotelian diagrams (e.g. the classical square, but also hexagons, octagons, etc.). Furthermore, these applications illustrate several key techniques and phe- nomena that are studied in logical geometry (e.g. Boolean closures, bitstring semantics, 1In a similar vein: “Ever since the beginning of the seventeenth century, almost every serious intellec- tual advance has had to begin with an attack on some Aristotelian doctrine; in logic, this is still true at the present day” (Russell, 1946, p. 237). For the role of Catholicism in the debate beween modern logic and Aristotelian syllogistics, see Jaspers and Seuren (2016). 2Although this specific type of educational context is probably less widespread now than it was a few decades ago, it still occurs quite frequently; see Demey (2017) for a partial explanation. 2 the interplay between Aristotelian and Boolean structure, logic-sensitivity, etc.). Finally, certain Aristotelian diagrams for TDD turn out to be intimately related to—and thus shed important new light on—other Aristotelian diagrams that are well-known in logical ge- ometry, but prima facie have nothing to do with definite descriptions. The paper is organized as follows. Section 2 provides a brief overview of Russell’s TDD and its later developments, focusing on those aspects that will be most relevant for our present purposes. Section 3 then introduces some of the key notions and techniques that are used in logical geometry to study Aristotelian diagrams. The next two sections develop and discuss various Aristotelian diagrams for TDD, and thus constitute the core of the paper. Section 4 first explains how TDD naturally gives rise to an Aristotelian square of opposition, and then shows that this square can be extended to an Aristotelian hexagon. Section 5 discusses the connection between the Aristotelian square for TDD and the traditional Aristotelian square for syllogistics. After arguing that the latter is already implicitly present in the former, both squares are integrated into a single Aris- totelian diagram, viz. an octagon, which is then shown to be closely related to other well-known Aristotelian diagrams. Section 6, finally, wraps things up and summarizes the main results obtained in this paper. 2 Russell’s Theory of Definite Descriptions In this section I will give a brief overview of Russell’s original TDD and some of its later developments. The focus will be on those aspects of TDD that are most relevant for the present paper, which means that some key topics—such as Donnellan’s (1966) distinction between referential and attributive uses of definite descriptions—will not be discussed in any detail.3 Definite descriptions are expressions of the form ‘the so-and-so’, such as ‘the Pres- ident of the United States’ and ‘the man standing over there’. They can occur in the subject position of a sentence (e.g. ‘the President of the United States is visiting France tomorrow’) as well as in the predicate position (e.g. ‘Barack Obama is the President of the United States’), but we will focus on the former. According to Russell’s TDD, a sentence of the form ‘the A is B’ can be translated into the language of first-order logic (FOL) as follows: 9xAx ^ 8y(Ay ! y = x) ^ Bx: For ease of notation, we will henceforth often make use of Neale’s (1990) restricted quantifier notation, and abbreviate ‘the A is B’ as [the x: Ax]Bx, which should be read as: ‘the unique x such that x is A, is B’. However, it should be emphasized that nothing of philosophical substance hinges on this notational convention. It can be shown that the first-order formalization of [the x: Ax]Bx stated above is equivalent to the conjunction of the following three formulas: (EX) 9xAx, (UN) 8x8y (Ax ^ Ay) ! x = y , (UV) 8x(Ax ! Bx). 3For more comprehensive overviews of the literature, see Neale (1990), Reimer and Bezuidenhout (2004) and Ludlow (2013); furthermore, see Elbourne (2013) for a recent alternative account. 3 These three formulas jointly express the truth conditions of [the x: Ax]Bx. First of all, the existence condition (EX) states that there exists at least one A; secondly, the unique- ness condition (UN) states that there exists at most one A; and finally, the universality condition (UV) states that all As are B.